<p>In the "Preparing for the ACT 2012-2013" booklet, I recently completed the Math section and had trouble with #38 on page 30. Could anyone help explain why the answer is G? Thanks. Here's the link:</p>
<p>Ok here’s an easy way to look at it. If you were to cut the white triangle (the one in the middle of the rectangle) in half horizontally, it would be the same as the triangles surrounding the white triangle. Now if you were to cut the white triangle vertically, it would be the same as the outer triangles. Then you would have 2 white triangles and 6 dark triangles which reduces to 1/3.</p>
<p>Here’s how I thought of it:</p>
<p>Consider triangle EFG. The ratio [EFG]:[EFBA] = 1/4 because triangle EFG has the same “base” as rectangle EFBA (namely side EF), 1/2 the height, divide by 2. Therefore, in rectangle EFBA, the ratio of the unshaded to the shaded region is 1:3.</p>
<p>By symmetry, the same applies for rectangle EFCD, and the answer is 1:3.</p>
<p>I thought about both those ways, but I wasn’t sure because all the triangles aren’t congruent, are they? They aren’t the same shape, but do they have the same area? That is what mainly tripped me up. I understand that the ratio of triangles is 1:3 in EFBA, but I didn’t know if I could assume that because I didn’t know if they were the same area.</p>
<p>You can assume they are the same because it states “points E and F are the midpointsof sides AD and BC of rectangle ABCD, point G is the intersection of AF and BE, and point H is the intersection of CE and DF”</p>
<p>You can easily show that [EFG] = [ABG], as those triangles are congruent ([x] denotes the area of x).</p>
<p>Also, you can show that [EFG] = [GFB]. Since G is the midpoint of BE, BG = EG. These two triangles share a common altitude (the segment from F to BE). Therefore triangles EFG and GFB have bases congruent to each other, and same height, so their areas are equal.</p>
<p>Can anyone explain how to do problem 56 of the 2012-2013 ACT test prep booklet?</p>
<p>Also, on problem 59, I see why 6 works for m, but why wouldn’t 3 also work, if n were 0 (an integer)?</p>
<p>Both questions are on page 33. <a href=“http://media.actstudent.org/documents/preparing.pdf[/url]”>http://media.actstudent.org/documents/preparing.pdf</a></p>
<ol>
<li><p>Note that the area of triangle ABC (denoted [ABC]) is [ABC] = (1/2) xy sin 70°. Also, [PQR] = (1/2) xy sin 110°. Since sin 70° = sin 110°, [ABC] = [PQR], so the answer is J, 30.</p></li>
<li><p>I think I posted the solution to the same question on another forum…anyway you know that x = -3 is the only possible solution for x. Since a quadratic always has two solutions in the complex plane, -3 must be a double root. Hence the polynomial is in the form (x - (-3))(x - (-3)) = (x+3)(x+3) = x^2 + 6x + 9. The leading coefficient is 1, which is what we want, so m = 6, C.</p></li>
</ol>
<p>rspence, can you elaborate more on #56? Why does sin70=sin110?</p>
<p>Because if you plot 70° and 110° on the unit circle, they have the same y-coordinates. Since sine is defined as the y-coordinate of a point on the unit circle with that terminal angle, sin 70° = sin 110°.</p>
<p>In general, sin x° = sin (180-x°).</p>
<p>Can someone please explain #47?</p>
<p>Number 47 is quite easy. It gives you one angle and since the lines are parallel, the other angle plus the given has to equal 180. So 180-82= 98. It says that bot of the lines AE and CE bisect the corresponding corners so all you have to do is divide the first given angles by 2 to get the interior angles. So 98/2= 49 and 82/2= 41. All you have to do is add them together and subtract that from 180. 49+41= 90 and 180-90= 90 so the answer is C.</p>
<p>Can someone explain 58? I have a program that does SAS, SSS, etc. It said the missing side was ~11.7… I thought the answer was B. Is the booklet wrong? My calculator always gives me the right answer on questions like these.</p>
<p>I think you mean 57. And the correct answer is E. I thought it was C as well but ive noticed whenever it gives you like the law of sines or cosines, you always use it. They dont usually put it there unless you absolutely need it because chances are you dont have it memorized.</p>
<p>Yeah, I think you meant 57. Applying the law of cosines yields E) as the correct answer.</p>
<p>oh alright, thanks.</p>